## Abstract and Applied Analysis

### Picard Successive Approximation Method for Solving Differential Equations Arising in Fractal Heat Transfer with Local Fractional Derivative

#### Abstract

The Fourier law of one-dimensional heat conduction equation in fractal media is investigated in this paper. An approximate solution to one-dimensional local fractional Volterra integral equation of the second kind, which is derived from the transformation of Fourier flux equation in discontinuous media, is considered. The Picard successive approximation method is applied to solve the temperature field based on the given Mittag-Leffler-type Fourier flux distribution in fractal media. The nondifferential approximate solutions are given to show the efficiency of the present method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 395710, 5 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425047794

Digital Object Identifier
doi:10.1155/2014/395710

Mathematical Reviews number (MathSciNet)
MR3170404

Zentralblatt MATH identifier
07022305

#### Citation

Yang, Ai-Min; Zhang, Cheng; Jafari, Hossein; Cattani, Carlo; Jiao, Ying. Picard Successive Approximation Method for Solving Differential Equations Arising in Fractal Heat Transfer with Local Fractional Derivative. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 395710, 5 pages. doi:10.1155/2014/395710. https://projecteuclid.org/euclid.aaa/1425047794

#### References

• J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 43, pp. 50–67, 1947.
• T. M. Shih, “A literature survey on numerical heat transfer,” Numerical Heat Transfer, vol. 5, no. 4, pp. 369–420, 1982.
• B. C. Choi and S. W. Churchill, “A technique for obtaining approximate solutions for a class of integral equations arising in radiative heat transfer,” International Journal of Heat and Fluid Flow, vol. 6, no. 1, pp. 42–48, 1985.
• Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” Journal of Thermal Stresses, vol. 28, no. 1, pp. 83–102, 2004.
• M. Dehghan, “The one-dimensional heat equation subject to a boundary integral specification,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 661–675, 2007.
• Y. Ioannou, M. M. Fyrillas, and C. Doumanidis, “Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer,” Engineering Analysis with Boundary Elements, vol. 36, no. 8, pp. 1278–1283, 2012.
• S. Nadeem and N. S. Akbar, “Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of adomian decomposition method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 3844–3855, 2009.
• A.-M. Wazwaz and M. S. Mehanna, “The combined Laplace-Adomian method for handling singular integral equation of heat transfer,” International Journal of Nonlinear Science, vol. 10, no. 2, pp. 248–252, 2010.
• S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109–113, 2006.
• B. Raftari and K. Vajravelu, “Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4149–4162, 2012.
• A. A. Joneidi, D. D. Ganji, and M. Babaelahi, “Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity,” International Communications in Heat and Mass Transfer, vol. 36, no. 7, pp. 757–762, 2009.
• C. Cattani and E. Laserra, “Spline-wavelets techniques for heat propagation,” Journal of Information & Optimization Sciences, vol. 24, no. 3, pp. 485–496, 2003.
• N. Simões, A. Tadeu, J. António, and W. Mansur, “Transient heat conduction under nonzero initial conditions: a solution using the boundary element method in the frequency domain,” Engineering Analysis with Boundary Elements, vol. 36, no. 4, pp. 562–567, 2012.
• J. Hristov, “Approximate solutions to fractional sub-diffusion equations: the heat-balance integral method,” The European Physical Journal, vol. 193, no. 1, pp. 229–243, 2011.
• J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,” Thermal Science, vol. 14, no. 2, pp. 291–316, 2010.
• D. D. Ganji and H. Sajjadi, “New analytical solution for natural convection of Darcian fluid in porous media prescribed surface heat flux,” Thermal Science, vol. 15, no. 2, pp. 221–227, 2011.
• X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013.
• X.-J. Yang, “Picard's approximation method for solving a class of local fractional Volterra integral equations,” Advances in Intelligent Transportation Systems, vol. 1, no. 3, pp. 67–70, 2012.
• R. R. Nigmatullin, “The realization of the generalized transfer equation in a medium with fractal geometry,” Physica Status Solidi B, vol. 133, no. 1, pp. 425–430, 1986.
• K. Davey and R. Prosser, “Analytical solutions for heat transfer on fractal and pre-fractal domains,” Applied Mathematical Modelling, vol. 37, no. 1-2, pp. 554–569, 2013.
• X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
• Y. Z. Zhang, A. M. Yang, and X.-J. Yang, “1-D heat conduction in a fractal medium: a solution by the local fractional Fourier series method,” Thermal Science, vol. 17, no. 3, pp. 953–956, 2013.
• M.-S. Hu, D. Baleanu, and X.-J. Yang, “One-phase problems for discontinuous heat transfer in fractal media,” Mathematical Problems in Engineering, vol. 2013, Article ID 358473, 3 pages, 2013.
• X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.
• M. Li and W. Zhao, “On bandlimitedness and lag-limitedness of fractional Gaussian noise,” Physica A, vol. 392, no. 9, pp. 1955–1961, 2013.
• M. Li, “Approximating ideal filters by systems of fractional order,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 365054, 6 pages, 2012.
• M. Li and W. Zhao, “On $1/f$ noise,” Mathematical Problems in Engineering, vol. 2012, Article ID 673648, 23 pages, 2012.
• M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and M. Li, “Chebyshev wavelets method for solution of nonlinear fractional integro-differential equations in a large interval,” Advances in Mathematical Physics, vol. 2013, Article ID 482083, 12 pages, 2013.
• S. S. Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329–336, 2006.
• Q. Wang, “Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006. \endinput